This invention is a method for the automatic conversion of a physical object into a 3D digital model. Combined with 3D scanning technologies it can reverse engineer mechanical, organic, and other shapes.
The reconstruction of 3D physical objects has applications in various commercial markets and has been practiced in the Arts and in Engineering disciplines. The computational problem of converting physical measurements into a full-fledged digital model has been studied in computer graphics and in computer-aided geometric design, both of which are disciplines within the Computer Sciences.
This invention concerns the automatic conversion of a 3-dimensional physical object into a digital model of exactly the same shape. The method is illustrated in FIG. 1 and the logical steps are depicted in FIG. 2. It starts with measurements taken with a 3D scanner. The measurements are points on the surface, which are automatically converted into a surface description using triangles connected to each other along shared edges. The surface description can either be used to produce a physical copy of the object using a 3D printing machine, and it can be further processed with animation or geometric design software.
The prior art uses a variety of systems to make 3D models of objects including manual and semi-manual solutions. The most frequently used method in the animation industry, where looks are more important than accuracy, is to use patches of spline surfaces which can be fit manually over clouds or sets of measured points. Semi-manual methods are common in the mechanical computer-aided design (CAD) industry where parameterized patches are fit over subsets of the measurements identified in a user-guided process. The commercial software CopyCAD by Delcam, Strim by Matra and Surfacer by Imageware are examples of this strategy.
Assuming the points are already connected to a surface by edges and triangles, there is a variety of methods available for replacing the piecewise linear description by a collection of curved spline patches. Charles Loop Smooth Spline Surfaces over Irregular Meshes (1994) and Jorg Peters, C1-Surface Splines (1995) decompose the triangles into smaller pieces and then replace each piece by a spline patches so the entire surface is continuous and differentiable. Both methods results in a large number of patches, which defeats the main purpose of introducing splines. Venkat Krishnamurthy and Marc Levoy, Fitting Smooth Surfaces to Dense Polygon Meshes (1996) address this shortcoming by manually decomposing the surface into regions and automatically fitting the corresponding spline patches using spring meshes. Matthias Eck and Hugues Hoppe, Automatic Reconstruction of B-Spline Surfaces or Arbitrary Topological Type (1996) automate the entire patch fitting process by first decimating the triangulated surface and then fitting parametrized patches using regression methods. Similarly, Chandrajit L. Bajaj, Jindon Chen and Guoliang Xu, Modeling with Cubic A-Patches (1995) use regression to fit implicit patches over pieces of a triangulated surface. While the latter two methods are automatic in fitting patches, they do not automate the task of shape reconstruction, which is needed to produce the triangulated surface and is a prerequisite of the patch fitting methods.
Among the automatic solutions, three approaches are distinguished: reconstruction from slices, reconstruction from dense measurement, and reconstruction from 3D complexes. The first two approaches are limited to shapes that can be described by a closed surface.
The reconstruction of a surface is considerably easier than in the general case if the measured points represent a sequence of parallel slices. Henry Fuchs, Zvi M. Kedem and Sam P. Uselton, Optional Surface Reconstruction from Planar Contours (1977) show how to connect two polygons in parallel planes with a cylindrical surface of minimum area. The problem is more difficult if there are more than one polygon per slice. Various heuristics for determining which polygons to connect and how to connect them have been investigated. For example, David Meyers, Shelly Skinner and Kenneth Sloan, Surfaces from Contours (1992) use the minimum set of edges that connect all points, known as a spanning tree of the measurements to guide the matching process for the polygons. Jean-Daniel Boissonnat, Shape Reconstruction from Planar Cross Sections (1988) uses the 3D Delaunay complex for both purposes: to decide which polygons to connect and also how to connect them. In spite of the effort reflected in a large body of literature, no algorithm appears to produce surfaces from sliced data in a generally satisfactory manner to produce a 3D model. Nevertheless, the reconstruction from slices is a fast and effective method for simple organic forms, such as eggs, bones, etc. They are part of commercially available systems such as the Cyberware scanners and medical imaging hardware and software.
A method for surface reconstruction that has become popular in the computer graphics community uses the measurements to construct a continuous function defined over the entire three-dimensional space. The surface is reconstructed as the set where the function assumes a constant value. This is the 2-dimensional analogue of the 1-dimensional contour or isoline in geographic maps that traces out a curve along which a possibly mountainous landscape assumes a constant height. The 2-dimensional analogue of an isoline is an isosurface and can be constructed with the marching cube algorithm. Hugues Hoppe, Tony DeRose, Tom Duchamp, John McDonaLD, Werner Stuezle, Surface Reconstruction from Unorganized Points (1992) construct this function so it approximates the signed Euclidean distance from the measured surface. A necessary assumption in their work is that measurement are uniformly dense over the entire surface and that the density exceeds the smallest feature size of the shape. Brian Curless and Marc Levoy, A Volumetric Method for Building Complex Models from Range Images (1996) use information about rays available from some types of scanners to extrapolated the function over gaps in the measured data. A fundamental requirement of this approach is that the signed distance function is differentiable in the normal direction along the entire surface, which is not possible unless the surface is a closed manifold. In other words, the surface is restricted to form the boundary of a solid volume in space. Examples of surfaces that do not satisfy this property are thin sheets or the common case where only a portion of the volume""s surface is accessible to measurement.
A 3D complex decomposes a 3-dimensional volume into cells. An example is the Delaunay complex of a point set, which connects the points with edges and triangles and this way decomposes the convex hull of the given points into tetrahedra. Except for Remco Veltkamp, Closed Object Boundaries from Scattered Points (1994) all work on shape reconstruction via 3D complexes is based on the Delaunay complex A representation of the shape is extracted from the complex by selecting an appropriate set of triangles, edges, and vertices. The various approaches differ in their selection process.
Jean-Daniel Boissonnat, Geometric Structures for Three-Dimensional Shape Representation (1984) describes this approach in general terms and gives some heuristics that sculpt a shape by removing tetrahedra from outside in. The weakness of the method is the lack of an effective rule for deciding which tetrahedra to remove and in what sequence. Herbert Edelsbrunner and Ernst Mucke, Three-Dimensional Alpha Shapes (1994) extend the concept of alpha shapes from 2D to 3D and define them as subcomplexes of the Delaunay complex. They give a rule when a tetrahedron, triangle, edge, vertex belongs to the alpha shape. This rule is exclusively expressed in terms of distance relationship, and it succeeds in reconstructing a shape provided the measured data points are uniformly distributed over its surface and possibly its interior: Chandrajit Bajaj, Fausto Bernardini and Guoliang Xu, Automatic Reconstruction of Surfaces and Scalar Fields from 3D Scans (1995) exploit alpha shapes for that purpose and report limitations resulting from this requirement.
The shape reconstruction part of this invention, referred to as wrap process, overcomes the past limitations. It also uses the Delaunay complex, but differs from all above methods by an effective selection rule that is unambiguous and does not depend on the uniformity and density of the point distribution.
The differences between the invention described in this application and other methods can be grouped into primarily theoretical, practical, or paradigmatic. The most important theoretical differences to other shape reconstruction methods are that the wrap process (I) can deal with any set of 3D point measurements, and (II) uses rational and justified rules.
Both theoretical differences have important practical and also paradigmatic consequences. Point (I) refers to the fact that most earlier methods limit the set of measurements to certain types in order to eliminate ambiguities in the reconstruction of the shape. The wrap process includes global topological considerations that disambiguate every set of measurements. Point (II) refers to the fact that the surface reconstructed by the wrap process can be defined, and the wrap process is merely an execution of that definition. All other reconstruction methods, with the exception of alpha shapes, leave some ambiguity to be resolved by the implementation. It is therefore not possible to predict the exact shape without executing the implementations.
The three most important practical advantages of the wrap process are direct consequences of the theoretical differences just mentioned:
(I) the wrap process can work in connection with any 3D scanner,
(II) the reconstruction is fully automatic,
(III) the resulting shape has a representation that permits editing, if desired.
Point (I) says that the invention provides a standard and universal interface to 3D point input devices. Current 3D scanners differ dramatically in quantity and quality of the 3D measurements, and they also differ in the kind of information they provide. As a lowest common denominator, each measurement is reported with enough information to unambiguously determine its location in space, e.g. in terms of Cartesian coordinates. Sometimes the measurement includes a half-line along which the measurement was made, or the color and texture of the object at the measured location. The quantity of measured points depends on the technology and ranges from just a few points per second for touch-probe devices to a few hundred thousand points per second for large laser range equipment.
Point (II) asserts that the wrap process is fully automatic, which is a decisive factor in many commercial applications, such as software used inside 3D copiers, 3D printers and 3D fax machines. Point (III) refers to the fact that the resulting shape can still be edited. This does not contradict the claim that the reconstruction is automatic but rather refers to imperfections due to incomplete and noisy measurements, and to situations where the original model is to be modified, for example to make small variations of the same design.
The invention described implies a major improvement of the physical design paradigm, which operates through copying physical objects into the digital domain. It is generally complementary to conceptual design paradigm, which provides tools to manually generate digital models on the computer.
In manufacturing, physical design is often referred to as reverse engineering since shape information is copied rather than created from engineering drawings. A major innovation brought about by this invention is that the process is now automatic. Another difference is how the physical design process is implemented results from an improvement of the basic invention, where the shape is reconstructed incrementally. Because the shape is uniquely defined for any set of measurements, also the changes necessary to reflect new and additional measurements are unambiguously defined. It is therefore possible to run the measuring and reconstruction steps of the entire process in an interleaved fashion.
An important practical consequence of this novel possibility is that large sets of measurements can now be filtered. All measurements are read and considered, but only the ones that makes a contribution and differences in the reconstructed shape are incorporated into the digital representation. This alleviates the currently most severe limitation shared by all shape reconstruction methods, which is the inability to digest large sets of measured data due to the limited availability of hardware memory.